Degree Conditions for Completely Independent Spanning Trees of Bipartite Graphs

Abstract

Let $T_1,T_2,\ldots , T_k$ be $k(\ge 2)$ spanning trees of a graph G. The trees $T_1,T_2,\ldots , T_k$ are completely independent if the paths connecting any two vertices of G in these k trees are pairwise internally disjoint. Sufficient conditions for a graph to possess k completely independent spanning trees have attracted popular attention intensively. In this paper, we focus on such sufficient conditions for bipartite graphs, and show that a bipartite graph $G=(X\cup Y, E)$ with order $\nu =m+n,$ $m=|X|\ge 2k$ and $n=|Y|\ge m,$ has k completely independent spanning trees if for every pair of vertices $x\in X,y\in Y,d(x)\ge \frac{(k-1)n}{k}+2,d(y)\ge \frac{(k-1)m}{k}+2.$ Furthermore, we obtain that a bipartite graph $G=(X\cup Y, E)$ with order $\nu \ge 8, |X|\ge 2k$ and $|Y|\ge 2k,$ has k completely independent spanning trees if the minimum degree $\delta (G)\ge \lfloor \frac{(3\times 2^{k-2}-1)\nu }{3\times 2^{k-1}}\rfloor +2,$ or $\delta (G)\ge \frac{(k-1)\nu }{2k}+2$ and $|X|=|Y|.$